On the differentiability class of the admissible square roots of regular nonnegative functions

We investigate the possibility of writing $f = g^2$ when $f$ is a $C^k$ nonnegative function with $k \geq 6$. We prove that, assuming that $f$ vanishes at all its local minima, it is possible to get $g \in C^2$ and three times differentiable at every point, but that one cannot ensure any additional regularity.